Dear All,
following some feedback from Bruno Tremblay, on behalf of SIMIP, I'd like to modify the standard names proposed for sea ice stress and strain invariants last month (25th April).
The proposed terms are refer to the first and second invariants of the horizontal stress and rate of strain tensors in the sea ice. I had not appreciated that the terms "first invariant" and "second invariant" are rather more flexible than we want for a standard name. The fist invariant, for instance, is sometimes represented by the trace of the tensor, and sometime by half the trace. These two quantities clearly carry the same information, but we need to know which one. In the SIMIP request they want half the trace of the stress tensor, because that is the form most commonly used in the sea ice community. This quantity is generally referred to as the "average normal stress" (the normal stress terms being the two diagonal elements of the stress tensor).
Similarly, the second invariants requested are not, as I had assumed, the effective stress/rate of strain (which is the square root of the determinant) but the "maximum shear stress/rate of strain", which is the maximum value of the off-diagonal tensor element, maximized over all coordinate rotations. At first I thought this was a completely different quantity, but, following Bruno's patient explanations, I now understand that it is related to the first invariant and the determinant by a simple algebraic equation, and hence represents the 2nd invariant just as well as the determinant or effective stress.
We only have requests for terms related to the horizontal stress/rate of strain, but I feel that the standard names should be explicit about this restriction to the horizontal plane, to avoid ambiguity.
Taking this into account, I'd like to propose the following modified names:
(3a) sishevel Maximum shear strain rate of sea-ice velocity field (s-1)
Maximum shear strain rate of sea-ice velocity field (second shear strain invariant: maximum is taken over coordinate rotations)
sea_ice_horizontal_shear_strain_rate_maximum_over_coordinate_rotation
Help text: "Stress is the symmetric component of the tensor representing the gradient of internal forces (e.g. in ice). Shear stress refers to off-diagonal element(s) of the stress tensor (a single element for horizontal shear stress). The maximum over coordinate rotations of the shear strain rate, often referred to as the maximum shear strain [rate], represents the second invariant of strain rate."
(3b) sistremax Maximum shear stress in sea ice (N m-1)
Maximum shear stress in sea ice (second stress invariant)
sea_ice_horizontal_shear_stress_maximum_over_coordinate_rotation
Help text: "Stress is the symmetric component of the tensor representing the gradient of internal forces (e.g. in ice). Shear stress refers to off-diagonal element(s) of the stress tensor (a single element for horizontal shear stress). The maximum over coordinate rotations of the shear stress, often referred to as the maximum shear stress, represents the second invariant of stress."
(3c) sistresave Average normal stress in sea ice (N m-1)
Average normal stress in sea ice (first stress invariant: average of diagonal elements of the stress tensor)
sea_ice_average_normal_horizontal_stress
Help text: "Stress is the symmetric component of the tensor representing the gradient of internal forces (e.g. in ice). Horizontal stress refers to the stress in the horizontal plane. Average normal stress refers to the average of the diagonal elements of the stress tensor and represents the first invariant of stress."
regards,
Martin
________________________________
From: Bruno Tremblay <bruno.tremblay at mcgill.ca>
Sent: 30 May 2018 17:27
To: Juckes, Martin (STFC,RAL,RALSP)
Cc: Dirk Notz; Pamment, Alison (STFC,RAL,RALSP)
Subject: Re: SIMIP -- a few remaining issues
Hi Martin
What you write is entirely correct. There is a factor of 2 difference. But since a constant factor in front of the invariant does not change the invariant nature we opt for a definition that is directly linked with the the way we plot the yield curve and write the constitutive relation.
I am glad it helped
Bruno
On Wed, May 30, 2018 at 11:37 AM, Martin Juckes - UKRI STFC <martin.juckes at stfc.ac.uk<mailto:martin.juckes at stfc.ac.uk>> wrote:
Dear Bruno,
thanks, that clears up a lot.
I'm afraid I'm still struggling with the relation between the "maximum shear stress" and the determinant of the stress tensor. You point out that the coordinates can be rotated to make the horizontal shear stress diagonal -- lets call the components A11 and A22. The determinant is then clearly det=A11*A22. If we rotate the coordinates by 45 degrees, I would expect the off-diagonal elements to be x=0.5*(A22-A11). If y=0.5*(A11+A22), then y**2 - x**2 = 2*det, so value maximum shear stress x is related to the determinant, but not exactly equal to it.
If these are just different, inter-changeable, approaches to representing invariant information, then we just need to be clear about this in the definitions.
regards,
Martin
________________________________
From: Bruno Tremblay <bruno.tremblay at mcgill.ca<mailto:bruno.tremblay at mcgill.ca>>
Sent: 30 May 2018 15:38
To: Juckes, Martin (STFC,RAL,RALSP)
Cc: Dirk Notz; Pamment, Alison (STFC,RAL,RALSP)
Subject: Re: SIMIP -- a few remaining issues
Hi Martin et al
Please see below after BT>> for comments
I hope this helps
Bruno
On Wed, May 30, 2018 at 5:59 AM, Martin Juckes - UKRI STFC <martin.juckes at stfc.ac.uk<mailto:martin.juckes at stfc.ac.uk><mailto:martin.juckes at stfc.ac.uk<mailto:martin.juckes at stfc.ac.uk>>> wrote:
Dear Dirk, Bruno,
thanks for the feedback.
Firstly, we do need to be clear about which variables are requested. At the moment we have sidivvel, sishevel, sistremax and sistresave requested.
'sidivvel' has a corresponding CF standard name "divergence_of_sea_ice_velocity" and a data request description "Divergence of sea-ice velocity field (first shear strain invariant)".
BT>> There is an inconsistency here. Strain has units of m/m (or unitless) and is the % elongation of a linear elastic material when compressed, pulled or sheared. This is used in linear elastic solid sea ice models such as the Elasto Brittle model but even there, people take the derivative of sigma with respect to time so that the d(sigma)/dt is expressed in term of the strain rates rather than the strain.
Bottom line:
1- We are speaking here of the first shear strain rate (m/s / m or sec^-1) invariant, not the first strain invariant. Many refer to it as the strain invariant but this is incorrect.
2- I said yesterday, we did not request to have the divergence of the sea ice velocity. Please scratch that. The first strain rate invariant is eps_11 + eps_22 which is the divergence of the sea ice velocity since eps_11 = du/dx and eps_22 = dv/dy.
The variable list provided by Dirk pairs sidivvel with sishevel, "Maximum shear of sea-ice velocity field (second shear strain invariant)". Both variables are requested with units of "s-1", which perhaps implies that they are related to the rate of strain, rather than strain itself, which would be dimensionless? If sishevel is not related to the rate of strain of the sea-ice velocity field (i.e. the symmetric component of the gradient of the velocity field), please clarify what it is.
BT>> Correct, everything in sea ice models is expressed in terms of strain rates rather than strain. Many use the term strain, when they really mean strain rate. It is the maximum shear of sea ice velocity field expressed in terms of the strain rate.
There appears to be some relation between sidivvel, which is referred to as the "first shear strain invariant", and sistreave, which is referred to as the "first stress invariant". And yet, if we have a tensor with elements a11, a12, a21, a22, then it is clear that "divergence" would be a11 + a22, and "average normal ...." would be 0.5*(a11+a22). We need a more precise definition -- do you want the average of the diagonal elements of the tensor or the sum?
BT>>
sidivvel is the first strain rate invariant, not the first shear strain invariant. The second strain rate invariant that is the maximum shear strain rate.
We want the sum for the first strain rate invariant (du/dv + dv/dy) and the average for the first stress invariant because the bulk viscosity in the sea ice constitutive relation is multiplied by the divergence of sea ice velocity (i.e. the sum) and because when we plot the stress in stress invariant space (to get the yield curve), we use the average normal stress (not the sum of the normal stresses). We could give the sum for both as long as we are clear in the definition with words.
Within the data request we have variable labels, titles, descriptions and standard names. The titles and descriptions often reflect the wording in the standard name, but sometimes the titles and descriptions are adjusted to reflect more of the usage in the community requesting the diagnostic, while the standard name needs to follow usage patterns established within the CF convention. The breadth of the CF convention means that these patterns of usage may look odd, but that is an unavoidable consequence of using such a standard: the advantage of having a common approach are huge.
The terms "maximum shear stress" and "maximum shear strain rate" may be unambiguous within the sea ice community, but there is an element of jargon here. The phrase "maximum shear strain" appears to be more widely used to refer to a critical value of shear strain which a material can withstand before breaking, but I think that a different interpretation is wanted here. The
BT>> No. The yield curve is expressed in terms of the stresses not strain rate. I.e. in your sentence above, you should write " ...to refer to a critical value of shear stress that a material can withstand before breaking.
maximum is, I believe, being taken over rotations of the horizontal stress and strain rate tensors. Is that correct?
BT>> Yes correct. A 2D symmetric tensor can always be rotated in such a way that the off-diagonal components will be zero. The normal components are then the principal stresses (the maximum and minimum axial stresses) and they are also the eigen-values of the 2D stress tensor. The amount of rotation that one must impose to get to the principal axes of stress is found from the rotation matrix composed of the two eigen-vectors associated with the two eigen-values. If we rotate the principal axis system by 45 degrees, then the off-diagonal components of the 2D stress tensor is the 2nd stress invariant (or the maximum shear stress) and the diagonal components are equal to each other and they are the average nornal stress (or the first stress invariant). These are two sets of stress invariants thatt are used in the community, one is called the principal stress and the other are called the stress invariants.
So a well formed standard name might be "sea_ice_velocity_horizontal_shear_strain_rate_maximum_over_coordinate_rotations". i.e. this is a diagnostic based on the horizontal strain rate tensor of the sea ice velocity field, and it is defined as the maximum value of the horizontal shear strain rate wrt. coordinate rotations. Besides commenting on the choice of CF terminology, could you also comment on the definition.
BT>> Yes this is a wordy definition but if you have the space then it is entirely correct.
I'm puzzled by the connection between the "2nd invariant of stress" and "maximum shear stress": I've found multiple sources identifying the 2nd invariant of a rank two tensor with the determinant, but none that relate it to a maximum over coordinate rotations.
BT>> They are exactly the same. When we rotate a 2D symmetric matrix, we can write the off-diagonal component and normal component using parametric equations of a circle. This circle is called the Mohr-Coulomb circle of stress. The center of the circle is the average normal stress (or first stress invariant ) and the radius of the circle is the maximum shear stress (or the second stress invariant).
BT>>
regards,
Martin
________________________________
From: Dirk Notz <dirk.notz at mpimet.mpg.de<mailto:dirk.notz at mpimet.mpg.de><mailto:dirk.notz at mpimet.mpg.de<mailto:dirk.notz at mpimet.mpg.de>>>
Sent: 30 May 2018 08:02
To: Juckes, Martin (STFC,RAL,RALSP); Pamment, Alison (STFC,RAL,RALSP)
Subject: Re: SIMIP -- a few remaining issues
Dear Martin,
please find below, after the excerpt from your initial email, some
thoughts from our sea-ice dynamics expert Bruno Tremblay. I generally
share his view, but I know that naming conventions are a difficult and
sometimes non-intuitive issue. I believe that you and Alison are better
qualified than us to judge the relevance of these thoughts.
Thanks,
Dirk
> (3) Stresses and strains
>
> --------------------------------
>
>
>
> There are 3 stress/strain variables (see below) for which I think we need a bit more information. Presumably these refer to the horizontal stress and strain fields? Since these are rank two tensors, the first invariant will be the trace, or sum of eigenvalues, and the 2nd will be the determinant, or product of eigenvalues. Given the units, it looks as though you are interpreting the 2nd invariant as the square root of the determinant? I found one source which suggest that the square root of the 2nd invariant is often used instead of the invariant itself, and referred to as the "effective stress" (e.g. http://www.engr.colostate.edu/~thompson/hPage/CourseMat/Tutorials/Solid_Mechanics/J2.pdf -- effective stress = square root [ 3 * 2nd invariant] ), but I don't know how wide spread this usage is.
>
>
> We don't have any standard names referring to the invariants of tensors, so we need some new terminology here.
>
>
> (3a) sishevel Maximum shear of sea-ice velocity field (s-1)
>
> Maximum shear of sea-ice velocity field (second shear strain invariant)
>
>
> sea_ice_second_invariant_of_horizontal_strain_tensor[_expressed_as_effective_strain]
>
> The last part, "_expressed_as_effective_stress" may be needed to deal with the fact that the second invariant would usually have units s-2.
>
>
> (3b) sistremax Maximum shear stress in sea ice (N m-1)
> Maximum shear stress in sea ice (second stress invariant)
>
> sea_ice_second_invariant_of_horizontal_stress_tensor[_expressed_as_effective_stress]
>
> (3c) sistresave Average normal stress in sea ice (N m-1)
> Average normal stress in sea ice (first stress invariant)
>
> sea_ice_trace_of_horizontal_stress_tensor
> Here we can just refer to the trace, but the help text could also refer to the "first invariant".
Reply from Bruno Tremblay
[...]
There are two strain invariants and two stress invariants. I am not sure
the comment about maximum shear in the sea ice velocity field applies to
us. I don't recall requesting this.
The naming convention proposed by Martin are more mathematical. I lean
more towards names that refer to their physical meaning.
1st invariant of stress: AverageNormalStress
2nd invariant of stress: MaximumShearStress
1st invariant of strain: Divergence
2st invariant of strain: MaximumSheaStrainRate
This is the way I would call them.
I would stay away from the more mathematical definition since the trace
and the determinant (1st and 2nd invariants) or any other linear
combination of the trace and determinant are also invariant. The reason
we use the one defined above is because they have physical meaning. I.e.
ice will fail in shear when the maximum shear stress reaches the
critical shear strength of the ice, etc.
I hope this helps
Bruno
Received on Thu May 31 2018 - 02:54:54 BST
